I know nothing about stochastics, and this has been a bit hard to wrap my head around. What are their processes and purposes?
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In the board game Monopoly, if you know the current location of a piece, you can calculate the different spaces that piece could land after its next move, and the probability of landing on each of those spaces. Then, if you do the same calculation for each of those spaces, you can calculate probabilities for where the piece will be after two turns… then three turns… then 50 turns… etc. This system is a Markov chain: by breaking the process down into independent steps and using some math tricks, you can learn things about the entire system. For example, you can learn the most-visited spaces overall, or the probability of landing on a given space sometime in the next 5 turns.
It turns out that a lot of real-world problems can be broken down into individual random steps. For example, I use Markov chains in my research about how electrons move through special plastics. Based on where an electron is now, we can estimate where it will be after a nanosecond, or what paths it might take through the material.
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